PARTS OF NATURAL NUMBERS 1

In this chapter we will learn to speak the language of arithmetic. This will allow us to relate any two numbers. We will learn to say, for example, that 6 is the fourth part, or one quarter, of 24, and that 18 is three quarters of 24. We are not concerned here with "How do you do it?" but, rather, "What does that mean?"

What is more, to really understand Percent, it is necessary to understand parts, because a percent is a part of 100%. 50% means half -- because 50 is half of 100. 25% means a quarter beause 25 is a quarter of 100. And 20% means a fifth because 20 is the fifth part of 100.

1, 2, 3, 4, and so on, are the familiar numerals for the natural numbers. We calculate with those symbols, and so it has become conventional to call the symbols themselves "numbers." Yet a symbol is not what it symbolizes, what it stands for, which in this case is a number of discrete units.

By a number in what follows, we will mean a natural number.

Cardinal and ordinal

The names of the natural numbers have two forms: cardinal and ordinal. The cardinal forms are

One, two, three, four, and so on.

They answer the question How many? The ordinal forms are

First, second, third, fourth, and so on.

They answer the question Which one? We will see that those ordinal numbers express division into equal parts. With the exception of "half," an ordinal number will name which part -- the third part, the fourth, the fifth, and so on.

To each cardinal number except 1, there will correspond the name of a part.

2

Half

3

Third

4

Fourth

5

Fifth

And so on.

2.

What do we mean by the multiples of a number?

They are the numbers produced when that number is repeatedly added.

Here are the first few multiples of 5:

5, 10, 15, 20, 25, 30, 35, 40.

5 is the first multiple of 5; 10 is the second multiple; 15, the third; and so on.

3.

What does it mean to say that a smaller number is a part of a larger number?

It means that the larger number is a multiple of the smaller. Equivalently, the smaller is contained in the larger an exact number of times.

5, then, will be a certain part of each one of its multiples.

5, 10, 15, 20, 25, 30, 35, 40.

Since 15 is the third multiple of 5, we say that 5 is the third part of 15.

We use that same ordinal number to name the part.

Since 20 is the fourth multiple of 5, we call 5 the fourth part of 20. 5 is the fifth part of 25, the sixth part of 30; and so on. But, 5 is half of 10. (We do not say the second part.) And 5 is not a part of itself; there is no such thing as the first part.

So with the exception of "half," an ordinal number names into which parts a number has been divided.

15 has been divided into Thirds; that is, into three equal parts. The third part of 15 is 5.

This use of ordinal numbers does not imply a sequence: the first part, the second, the third, and so on. See below.

If we divide a number into four equal parts,

then we have divided it into fourths (or quarters); if into five equal parts,

into fifths. But if we divide into two equal parts, then we have divided it in half.

It is important to understand that we are not speaking here of proper fractions -- numbers that are less than 1 and that we need for measuring. We are explaining how the ordinal numbers --- third, fourth, fifth, etc. -- name the equal parts into which a number has been divided. When answering the questions of this Lesson, the student should not write fractions. We will come to those symbols in Lesson 20.

It should be clear that the ordinal names of the parts belong to language itself, and are prior to the names of the proper fractions, which are the parts of 1.

Why is the number we write as 1 over 3 -- - called "one-third"? Because the numerator is one third -- the third part -- of the denominator.

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That must be understood first. We can then explain that the number we call is the third part of 1.

In Lesson 4, we saw that when we divide by 10, we have taken 10% of the number. And when we divide by 100, we have taken 1%.

Therefore, 10% of $275 is $27.50. 1% is $2.75.

Note: Whenever we divide by any power of 10 -- the digits do not change.

275 ÷ 100 = 2.75

Conversely, then, if two numbers have the same digits, they differ by a power of 10.

Example 6. $85 is which part of $850?

Answer. Apart from the 0 at the end of $850, those numbers have the same digits. Therefore, they differ by a power of 10. 850 is in fact 10 times 85. (Lesson 4, Question 1.) Therefore, $85 is the tenth part of $850. To say the same thing, $85 is 10% of $850.

Example 7. $.98 is which part of $98?

Answer. They have the same digits. They differ by a power of 10.

$.98 is the hundredth part of $98. It is 1% of it.

Divisors

We say that a smaller number is a divisor of a larger number if the larger is a multiple of the smaller. 3 is a divisor of 12 because 12 is a multiple of 3. 5 is not a divisor of 12. We say however that a number is a divisor of itself. With the exception of the number itself, the divisors of a number are the only parts that a number has.

3 is the fourth part of 12. 6 is half of 12. 5 is not any part of 12. You cannot divide 12 people into groups of 5.

Example 8. Find all the divisors of 30 in pairs. Each divisor (except 30) is which part of 30?

Answer. Here are all the divisors of 30 in pairs:

1 and 30. (Because 1 × 30 = 30.)

2 and 15. (Because 2 × 15 = 30.)

3 and 10. (Because 3 × 10 = 30.)

5 and 6. (Because 5 × 6 = 30.)

On naming which part of 30, each divisor will say the ordinal name of its partner:

Every time we take half of something, we get twice as many parts. Half of a whole -- --

-- results in two equal parts. Each part is Half.

If we divide each Half in half --

-- the whole will be in four equal parts, or Quarters.

If we divide each Quarter in half --

-- the whole will then be in twice as many, that is, eight equal parts, or Eighths.

Half of an Eighth is a Sixteenth. Half of a Sixteenth is a Thirty-second. And so on.

Now, here are the number of equal parts that result when we repeatedly take half:

2, 4, 8, 16, 32, 64, and so on.

Those numbers are called the powers of 2. Repeated division in half is very common. The student should know the names of the sequence of those parts:

Halves, Quarters, Eighths, Sixteenths, Thirty-seconds, and so on.

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When we say "5 is the third part of 15," we do not imply a sequence: the first part, the second part, the third, and so on. When we speak of the third part, that is a different meaning for the word "third." It means each one of three equal parts that together make up the whole.

We say that we have divided 15 into thirds.

Yet "third" still retains its ordinal character. Because to the question, "Which part of 15 is 5?", we answer, "The third part." We use that ordinal number because 15 is the third multiple of 5.